In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number . p-variation is a measure of the regularity or smoothness of a function. Specifically, if , where is a metric space and I a totally ordered set, its p-variation is:
where D ranges over all finite partitions of the interval I.
The p variation of a function decreases with p. If f has finite p-variation and g is an ñ-Hölder continuous function, then has finite -variation.
The case when p is one is called total variation, and functions with a finite 1-variation are called bounded variation functions.
This concept should not be confused with the notion of p-th variation along a sequence of partitions, which is computed as a limit along a given sequence of time partitions:
For example for p=2, this corresponds to the concept of quadratic variation, which is different from 2-variation.
One can interpret the p-variation as a parameter-independent version of the Hölder norm, which also extends to discontinuous functions.
If f is ñ–Hölder continuous (i.e. its ñ–Hölder norm is finite) then its -variation is finite. Specifically, on an interval [a,b], .
If p is less than q then the space of functions of finite p-variation on a compact set is continuously embedded with norm 1 into those of finite q-variation. I.e. . However unlike the analogous situation with Hölder spaces the embedding is not compact. For example, consider the real functions on [0,1] given by . They are uniformly bounded in 1-variation and converge pointwise to a discontinuous function f but this not only is not a convergence in p-variation for any p but also is not uniform convergence.
If f and g are functions from [a, b] to with no common discontinuities and with f having finite p-variation and g having finite q-variation, with then the RiemannâÂÂStieltjes Integral
is well-defined. This integral is known as the Young integral because it comes from . The value of this definite integral is bounded by the Young-Loève estimate as follows
where C is a constant which only depends on p and q and þ is any number between a and b. If f and g are continuous, the indefinite integral is a continuous function with finite q-variation: If a ⤠s ⤠t ⤠b then , its q-variation on [s,t], is bounded by
where C is a constant which only depends on p and q.
A function from to e ÃÂ d real matrices is called an -valued one-form on .
If f is a Lipschitz continuous -valued one-form on , and X is a continuous function from the interval [a, b] to with finite p-variation with p less than 2, then the integral of f on X, , can be calculated because each component of f(X(t)) will be a path of finite p-variation and the integral is a sum of finitely many Young integrals. It provides the solution to the equation driven by the path X.
More significantly, if f is a Lipschitz continuous -valued one-form on , and X is a continuous function from the interval [a, b] to with finite p-variation with p less than 2, then Young integration is enough to establish the solution of the equation driven by the path X.
The theory of rough paths generalises the Young integral and Young differential equations and makes heavy use of the concept of p-variation.
p-variation should be contrasted with the quadratic variation which is used in stochastic analysis, which takes one stochastic process to another. In particular the definition of quadratic variation looks a bit like the definition of p-variation, when p has the value 2. Quadratic variation is defined as a limit as the partition gets finer, whereas p-variation is a supremum over all partitions. Thus the quadratic variation of a process could be smaller than its 2-variation. If W<sub>t</sub> is a standard Brownian motion on [0, T], then with probability one its p-variation is infinite for and finite otherwise. The quadratic variation of W is .
For a discrete time series of observations X<sub>0</sub>,...,X<sub>N</sub> it is straightforward to compute its p-variation with complexity of O(N<sup>2</sup>). Here is an example C++ code using dynamic programming:
There exist much more efficient, but also more complicated, algorithms for -valued processes
and for processes in arbitrary metric spaces.